Why vacuously true is defined for 'every' and is not defined for 'there exist' When get general logic statements of vacuous truth, seems that the allowed forms are only for 'all ($\forall$)', and not for 'there exist ($\exists$)'. For example, in wiki, it shows the possible universally quantified statements are:


*

*$\forall x:P(x)\Rightarrow Q(x)$, where it is the case that $\forall x:\neg P(x)$.

*$\forall x\in A:Q(x)$, where the set $A$ is empty.

*$\forall \xi :Q(\xi )$, where the symbol $\xi$ is restricted to a type that has no representatives. 


I dont understand why there is no form constructed by $\exists$, for example:

$\exists x\in A:Q(x)$, where the set $A$ is empty.

In my understanding, this statement is equivalent to say: "if there exist $x\in A$, then $Q(x)$ is true". When $A$ is empty, that means the "there exist $x\in A$" is false, i.e. the $P$ of "$P\Rightarrow Q$" is false, then logically, the statement should be vacuously true.
Could someone tell me where is wrong with this statement? 
(I know it's wrong as it is used to show "$\varnothing\not\subset A$ is false instead of vacuous true". See, for example, in discussion: "$X$ is not a subset of $A$", in symbols $X\not\subset A,$ means $\exists x(x\in X\text{ and }x\notin A).$ If $X$ has no elements, this existential statement is not true vacuously, it is simply false.).
 A: $\exists x \in A : Q(x)$ means there exists an $x$ that is a member of $A$ and $Q(x)$ is true.  If there are no members of $A$ this cannot be true.  We don't care about what $Q(x)$ says because $\exists x \in A$ is already false.
A: On $\exists x\in A:Q(x)$

In my understanding, this statement is equivalent to say: "if there exist $x\in A$, then $Q(x)$ is true"

Your understanding is wrong. Rather, the statement is ”There exists an $x$ such that $x\in A$ and $Q(x)$ is true.
Or in symbols, the long form for $\exists x\in A: Q(x)$ is not $\exists x: x\in A\implies Q(x)$, but rather $\exists x: x\in A\land Q(x)$.
Note that this is in contrast to $\forall x\in A: Q(x)$, which indeed does mean the same as $\forall x: x\in A\implies Q(x)$.
The rule for the $\exists$ quantifier can be derived from the rule of the $\forall$ quantifier as follows:
$$\begin{aligned}
\exists x\in A: Q(x)
&\iff \lnot \forall x\in A: \lnot Q(x)\\
&\iff \lnot \forall x: x\in A \implies \lnot Q(x)\\
&\iff \lnot \forall x: \lnot x\in A \lor \lnot Q(x)\\
&\iff \lnot \forall x: \lnot (x\in A \land Q(x))\\
&\iff \exists x: x\in A \land Q(x)
\end{aligned}$$
A: At the request of the OP, let me expand on my comments.
It's always a good idea to start with an intuitive picture of what's going on (if such a thing is available). In this case, consider the natural-language examples: "There is a unicorn which is blue" (= $\exists x\in U: B(x)$) versus "Every unicorn is blue" (= $\forall x\in U: B(x)$). The former is false, since there are no unicorns at all, let alone blue ones; the latter is true (there isn't a counterexample!).
Now, let's dive a bit more into the claim that "$\forall x\in U: B(x)$" is true. Certainly if I said "All unicorns are blue," you would respond "But that's silly - there aren't any unicorns at all!" The important point, however, is that silliness is not falsehood. Statements like this - which begin with a quantification over the empty set - are silly, but true. To convince yourself of this, go back to the definition of "$\forall$": a statement of the form $$\forall x:\varphi(x)$$ is false if and only if there is a counterexample - namely, some $x$ for which $\varphi(x)$ fails. Similarly, a statement of the form $$\forall x\in A: \varphi(x)$$ is false if and only if there is a counterexample - namely, some $x$ in $A$ for which $\varphi(x)$ fails. Since there is no unicorn which is not blue, the statement "All unicorns are blue" is true, even though it's silly. 
Incidentally, by the same argument, the statement "All unicorns are not blue"is also true. This is in fact a way you can tell that a set is empty: if you prove "All $x\in A$ are $P$" and "All $x\in A$ are not $P$," you can conclude that $A$ is empty, since that's the only way this situation could possibly happen.
